Parameter Tuning of a Hybrid Treecode-FMM on GPUs Rio Yokota, Lorena - - PowerPoint PPT Presentation

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Parameter Tuning of a Hybrid Treecode-FMM on GPUs Rio Yokota, Lorena Barba Department of Mechanical Engineering, Boston University Saturday, June 4, 2011 Previous Calculations N=3x10 9 : 6 sec (Yokota & Barba) N=3x10 9 : 20 sec 40 TFlops

SLIDE 1

Parameter Tuning of a Hybrid Treecode-FMM on GPUs

Rio Yokota, Lorena Barba Department of Mechanical Engineering, Boston University

Saturday, June 4, 2011

SLIDE 2

Previous Calculations

Nagasaki University DEGIMA cluster GTX295 x 380 = 760 GPUs Theoretical Peak : 700 TFlops Power Consumption : 100 kW

Astrophysics Treecode 100 TFlops N=3x109 : 6 sec (Nitadori & Hamada) Turbulence FMM 40 TFlops N=3x109 : 20 sec (Yokota & Barba)

Saturday, June 4, 2011

SLIDE 3

Treecode & FMM

Saturday, June 4, 2011

SLIDE 4

Multipole expansion

＊ j

do i = 1,N ff = 0 do j = 1,N ff = ff+1/ ( x( i )-x( j ) ) end do f( i ) = ff end do do k = 1,p gg = 0 do j = 1,N gg = gg+( x( j ) - xs )**( k-1 ) end do g(k) = gg end do do i = 1,N ff = 0 do k = 1,p ff = ff+( x( i )-xs )**( -k )*g( k-1 ) end do end do

1 xi − xj =

1 xi − x∗ p−1

xj − x∗ xi − x∗ k =

p−1

(xi − x∗)−k−1   

(xj − x∗)k   

1 1 − t =

p−1

tk

1 xi − xj = 1 xi − x∗ 1

1 − xj−x∗

xi−x∗

and Taylor expansion

gives

Saturday, June 4, 2011

SLIDE 5

Error Control

Treecode FMM Complexity Error

O(θp) O(p3θ−3)

Saturday, June 4, 2011

SLIDE 6

Error Optimization

Optimize p Optimize θ Better on GPUs?

Saturday, June 4, 2011

SLIDE 7

Stack based hybrid treewalk

target source push pop push pop push

Saturday, June 4, 2011

SLIDE 8

Parameter study

Complexity Error

O(θp) O(p3θ−3)

Saturday, June 4, 2011

SLIDE 9

Optimum parameters

Saturday, June 4, 2011

SLIDE 10

Parallel calculation

Nagasaki University DEGIMA 760 GTX295 GPUs Peak : 0.7 PFlops Tokyo Institute of Technology TSUBAME 2.0 4224 M2050 GPUs Peak : 2.4 PFlops

Saturday, June 4, 2011

SLIDE 11

Strong Scaling

1 2 4 8 16 32 64 128 256 512 50 100 150 200 250 300 350 400 Nprocs time x Nprocs [s] 1 2 4 8 16 32 64 128 256 512 50 100 150 200 250 300 350 400 Nprocs time x Nprocs [s]

tree construction mpisendp2p mpisendm2l P2Pkernel P2Mkernel M2Mkernel M2Lkernel L2Lkernel L2Pkernel

DEGIMA TSUBAME 2.0

N=108

Saturday, June 4, 2011

SLIDE 12

Weak Scaling

! "# #$% #&!' & $ (& ($ #& #$ "& )*+,-./01/2.03-44-4/567849 :;+-/54-39 <03=>/-?=>*=@;0A BCC/-?=>*=@;0A C7D/30++*A;3=@;0A 678/30++*A;3=@;0A :.--/30A4@.*3@;0A

0.5 PFlops TSUBAME 2.0

Saturday, June 4, 2011

SLIDE 13

Summary & Outlook

1. The stack based treewalk enables a simple but effective hybridization of treecodes

and FMMs

3. More tests need to be performed in the higher accuracy range and the overall

performance must be compared to other treecodes and FMMs

2. The optimum p and are different on CPUs and GPUs, but this difference is small

θ

Saturday, June 4, 2011